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Weakly monotone averaging functions

chapter
posted on 2014-01-01, 00:00 authored by Tim WilkinTim Wilkin, Gleb BeliakovGleb Beliakov, T Calvo
Averaging behaviour of aggregation functions depends on the fundamental property of monotonicity with respect to all arguments. Unfortunately this is a limiting property that ensures that many important averaging functions are excluded from the theoretical framework. We propose a definition for weakly monotone averaging functions to encompass the averaging aggregation functions in a framework with many commonly used non-monotonic means. Weakly monotonic averages are robust to outliers and noise, making them extremely important in practical applications. We show that several robust estimators of location are actually weakly monotone and we provide sufficient conditions for weak monotonicity of the Lehmer and Gini means and some mixture functions. In particular we show that mixture functions with Gaussian kernels, which arise frequently in image and signal processing applications, are actually weakly monotonic averages. Our concept of weak monotonicity provides a sound theoretical and practical basis for understanding both monotone and non-monotone averaging functions within the same framework. This allows us to effectively relate these previously disparate areas of research and gain a deeper understanding of averaging aggregation methods.

History

Title of book

Information processing and management of uncertainty in knowledge-based systems : 15th International Conference, IPMU 2014, Montpellier, France, July 15-19, 2014

Volume

444 CCIS

Issue

PART 3

Chapter number

38

Pagination

364 - 373

Publisher

Springer

Place of publication

Cham, Switzerland

ISSN

1865-0929

ISBN-13

9783319088518

Language

eng

Publication classification

B Book chapter; B1 Book chapter

Copyright notice

2014, Springer

Extent

57

Editor/Contributor(s)

A Laurent, O Strauss, B Bouchon-Meunier, R Yager